Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics

This is a brief survey of applications of the harmonic superspace methods to the models of N=4 supersymmetric quantum mechanics (SQM). The main focus is on a recent progress in constructing SQM models with couplings to the background non-Abelian gauge fields. Besides reviewing and systemizing the re...

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Datum:	2011
1. Verfasser:	Ivanov, E.A.
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Veröffentlicht:	Інститут математики НАН України 2011
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1467762019-02-12T01:23:21Z Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics Ivanov, E.A. This is a brief survey of applications of the harmonic superspace methods to the models of N=4 supersymmetric quantum mechanics (SQM). The main focus is on a recent progress in constructing SQM models with couplings to the background non-Abelian gauge fields. Besides reviewing and systemizing the relevant results, we present some new examples and make clarifying comments. 2011 Article Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics / E.A. Ivanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 81T60; 81Q60 DOI:10.3842/SIGMA.2011.015 http://dspace.nbuv.gov.ua/handle/123456789/146776 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	This is a brief survey of applications of the harmonic superspace methods to the models of N=4 supersymmetric quantum mechanics (SQM). The main focus is on a recent progress in constructing SQM models with couplings to the background non-Abelian gauge fields. Besides reviewing and systemizing the relevant results, we present some new examples and make clarifying comments.
format	Article
author	Ivanov, E.A.
spellingShingle	Ivanov, E.A. Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Ivanov, E.A.
author_sort	Ivanov, E.A.
title	Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics
title_short	Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics
title_full	Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics
title_fullStr	Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics
title_full_unstemmed	Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics
title_sort	harmonic superfields in n=4 supersymmetric quantum mechanics
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/146776
citation_txt	Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics / E.A. Ivanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 31 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT ivanovea harmonicsuperfieldsinn4supersymmetricquantummechanics
first_indexed	2025-07-11T00:33:40Z
last_indexed	2025-07-11T00:33:40Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 015, 14 pages Harmonic Superfields in N = 4 Supersymmetric Quantum Mechanics? Evgeny A. IVANOV Bogoliubov Laboratory of Theoretical Physics, JINR, 141980, Dubna, Moscow Region, Russia E-mail: eivanov@theor.jinr.ru Received December 20, 2010, in final form February 03, 2011; Published online February 11, 2011 doi:10.3842/SIGMA.2011.015 Abstract. This is a brief survey of applications of the harmonic superspace methods to the models of N = 4 supersymmetric quantum mechanics (SQM). The main focus is on a recent progress in constructing SQM models with couplings to the background non-Abelian gauge fields. Besides reviewing and systemizing the relevant results, we present some new examples and make clarifying comments. Key words: supersymmetry; harmonic superspace; mechanics 2010 Mathematics Subject Classification: 81T60; 81Q60 1 Introduction Supersymmetric quantum mechanics (SQM) [1] is the simplest (d = 1) supersymmetric theory. It has plenty of applications in various domains. Some salient features of the SQM models and their uses were already discussed during this Benasque meeting. Let us recall some other ones: • SQM models are capable to catch characteristic properties of the higher-dimensional su- persymmetric theories via the dimensional reduction [2]; • They provide superextensions of integrable models like Calogero–Moser systems [3, 4] and Landau-type models [5]; • An extended supersymmetry in d = 1 is specific; it exhibits some features which are not shared by its d > 1 counterparts. These are the so-called automorphic dualities between various supermultiplets with the different off-shell contents [6], the existence of nonlinear “cousins” of off-shell linear multiplets [7, 8], and some other ones. An efficient tool to deal with extended supersymmetries in d > 1 is known to be the harmonic superspace (HSS) [9, 10]. The HSS approach allowed to construct, for the first time, an off- shell formulation of hypermultiplets in N = 2, d = 4 and N = 1, d = 6 supersymmetry, as well as a formulation of N = 4, d = 4 supersymmetric Yang–Mills theory with the maximal number N = 3 of off-shell supersymmetries (at cost of an infinite number of the auxiliary fields appearing in the harmonic expansions of the relevant superfields). Some further consequences of the HSS approach for the d > 1 supersymmetric theories are listed, e.g., in [10, 11]. A natural extension of the HSS approach was applying it to d = 1 supersymmetric theories, i.e. SQM models. An N = 4, d = 1 version of the N = 2, d = 4 HSS was worked out in [8]. It proved to be a powerful device of the N ≥ 4 SQM model-building, as well as of getting new insights into the structure of d = 1 supersymmetries and their representations. In particular, it allowed one to understand interrelations between various N = 4 SQM models via the manifestly ?This paper is a contribution to the Proceedings of the Workshop “Supersymmetric Quantum Me- chanics and Spectral Design” (July 18–30, 2010, Benasque, Spain). The full collection is available at http://www.emis.de/journals/SIGMA/SUSYQM2010.html mailto:eivanov@theor.jinr.ru http://dx.doi.org/10.3842/SIGMA.2011.015 http://www.emis.de/journals/SIGMA/SUSYQM2010.html 2 E.A. Ivanov N = 4 covariant gauging procedure [12, 13, 14]. As one more important application, it helped to construct new N = 4 superextensions of the Calogero-type models [4]. The latest developments of the d = 1 HSS approach concern applications in SQM models with the Lorentz-force type couplings to the external gauge fields, i.e. couplings of the form Am(x(t))ẋm(t). The major subject of this contribution is just a survey of these new applications from a common point of view, with some clarifying examples and further remarks. Let us adduce some reasons why SQM models with external gauge fields are of interest. One reason is that these models supply d = 1 prototypes of the p-branes world-volume cou- plings. The other one is the close relation of such models to supersymmetric versions of the Wilson loops and Berry phase (see, e.g., [15]). Also, they provide superextensions of the Landau problem and of the quantum Hall effect (see, e.g., [5]) and give quantum-mechanical realizations of Hopf maps (see, e.g., [16]). At last, they yield, as the particular “extreme” case, superexten- sions of the Chern–Simons mechanics [17]. Our consideration will be limited to the N = 4 SQM models with the background gauge field1. Until recently, only N = 4 superextensions of the couplings to Abelian background gauge fields were known. Their off-shell formulation within the N = 4, d = 1 HSS setting was given in the paper [8]. The coupling to non-Abelian gauge backgrounds was recently constructed in the papers [19, 20, 21] (see also [22, 23]). This construction essentially exploits the semi-dynamical (or spin, or isospin) supermultiplet (4,4,0) [4]. Bosonic fields of the spin multiplet are described by the U(1) gauged Wess–Zumino d = 1 action and, after quantization, yield generators of the gauge group SU(2). A key role is also played by the manifestly N = 4 supersymmetric d = 1 gauging procedure worked out in the papers [12, 13, 14]. It turns out that the requirement of off-shell N = 4 supersymmetry forces the external gauge potential to be (a) self-dual and (b) satisfying the 4D ’t Hooft ansatz or its 3D reduction. Moreover, in this case the gauge group should be SU(2). On the other hand, the on-shell N = 4 supersymmetry is compatible with the general self-dual background and an arbitrary gauge group [24, 19]. Most of the results reported here were obtained together with Francois Delduc, Sergey Fe- doruk, Maxim Konyushikhin, Olaf Lechtenfeld, Jiri Niederle2 and Andrei Smilga. 2 Harmonic N = 4, d = 1 superspace As a prerequisite to the main subject, let us recall the salient features of the d = 1 version of HSS. 2.1 From the ordinary N = 4, d = 1 superspace to the harmonic one The ordinary N = 4, d = 1 superspace is parametrized by the co-ordinates: (t, θα, θ̄α), α = 1, 2. Its harmonic extension is defined as: (t, θα, θ̄α) ⇒ (t, θα, θ̄α, u ± α ), u+αu−α = 1, u±α ∈ SU(2)Aut. (1) The main property of this d = 1 HSS is the existence of the so-called analytic basis in it3 (tA, θ +, θ̄+, u±α , θ −, θ̄−) ≡ (ζ, u±, θ−, θ̄−), θ± = θαu±α , θ̄ ± = θ̄αu±α , tA = t+ i(θ+θ̄− + θ−θ̄+). 1The on-shell N = 2 SQM models with couplings to a non-Abelian monopole background were considered, e.g., in [18]. 2Deceased. 3The original parametrization (1) will be referred to as the “central basis”. Harmonic Superfields in N = 4 Supersymmetric Quantum Mechanics 3 Passing to the analytic basis makes manifest the presence of the analytic subspace in the d = 1 HSS: (tA, θ +, θ̄+, u±α ) ≡ (ζ, u±) ⊂ (ζ, u±, θ−, θ̄−). It is closed by itself under the N = 4 supersymmetry, but has twice as less Grassmann coor- dinates compared to the full HSS. The superfields given on this subspace are called analytic superfields. They can be defined by the constraints which resemble the well known chirality condition: D+Φ = D̄+Φ = 0 ⇒ Φ = Φ(ζ, u±), D+ = ∂ ∂θ− , D̄+ = − ∂ ∂θ̄− . (2) An important ingredient of the HSS formalism is the harmonic derivatives, i.e. the derivatives with respect to the harmonic variables: D±± = u±α ∂ ∂u∓α + θ± ∂ ∂θ∓ + θ̄± ∂ ∂θ̄∓ + 2iθ±θ̄± ∂ ∂tA . The derivative D++ is distinguished in that it commutes with the spinor derivatives D+, D̄+. Then, if the superfield Φ is analytic, the superfield D++Φ is analytic as well: [D+, D++] = [D̄+, D++] = 0 ⇒ D++Φ(ζ, u±) is analytic, D+(D++Φ) = D̄+(D++Φ) = 0. The harmonic derivatives form an SU(2) algebra: [D++, D−−] = D0, D0 = u+ α ∂ ∂u+ α − u−α ∂ ∂u−α + θ+ ∂ ∂θ+ + θ̄+ ∂ ∂θ̄+ − θ− ∂ ∂θ− − θ̄− ∂ ∂θ̄− . The operator D0 counts the harmonic U(1) charge of the superfields given on the d = 1 HSS. It preserves the analyticity and is reduced to its pure harmonic part in the central basis. 2.2 Basic N = 4, d = 1 multiplets It turns out that the basic off-shell multiplets of N = 4, d = 1 supersymmetry are represented by analytic harmonic superfields subjected to the proper additional constraints. Below we briefly characterize these multiplets and quote their free superfield actions. We shall use for them the abbreviation (b,4,4− b), with b standing for the physical bosonic fields and 4− b for the auxiliary bosonic fields4. Depending on the choice of the action, some of the fields having a “physical” engineering dimension can become auxiliary, i.e. appear in the component action without time derivative on them. 1. (4, 4, 0) multiplet. The multiplet (4,4,0) is described by the superfield q+a(ζ, u) ∝ (xαa, χa, χ̄a), a = 1, 2, with the harmonic constraint D++q+a = 0, q+a = xαau+ α − 2θ+χa − 2θ̄+χ̄a − 2iθ+θ̄+ẋαau−α . (3) The free action of this multiplet reads Sfree(q) ∼ ∫ dtd4θdu q+aD−−q+ a ∼ ∫ dt (ẋαaẋαa + iχ̄aχ̇a) . (4) 4An additional set of admissible multiplets can be gained by changing the overall Grassmann parity of the relevant superfields. 4 E.A. Ivanov It also admits a Wess–Zumino (WZ) term which, in its simplest variant, is given by the analytic subspace action SWZ(q) ∼ ∫ dtAdudθ +dθ̄+C(ab)q +aq+b ∼ ∫ dtC(ab) ( ixαaẋα b + 4χaχ̄b ) . (5) 2. (3, 4, 1) multiplet. The multiplet (3,4,1) is described by the superfield L++(ζ, u) ∝ (`(αβ), ψα, ψ̄α, F ) which is subjected to the constraint D++L++ = 0, L++ = `αβu+ αu + β + i(θ+χα + θ̄+χ̄α)u+ α + θ+θ̄+(F − 2i ˙̀αβu+ αu − β ). The free action is: Sfree(`) ∼ ∫ dtd4θduL++(D−−)2L++ ∼ ∫ dt [( ˙̀αβ ˙̀ αβ − 1 2 F 2 ) + iψ̄αψ̇α ] . This multiplet also admits WZ-type N = 4 superfield invariants. 3. Gauge multiplet. An important multiplet is the gauge one described by an uncon- strained analytic superfield V ++(ζ, u), which exhibits the gauge freedom with an analytic su- perfield parameter: V ++′ = V ++ +D++Λ, Λ = Λ(ζ, u). This gauge freedom can be fixed so as to bring V ++ into the Wess–Zumino gauge with only one component B(t) (d = 1 “gauge field”): V ++ WZ = 2iθ+θ̄+B(t), δB = λ̇(t). 4. Gauged (4, 4, 0) multiplet. Using the superfield V ++, one can define a gauged version of the multiplet (4,4,0). It is represented by the superfields (v+, v̄+), v+′ = eiΛv+, v̄+′ = e−iΛv̄+, obeying a gauge-covariant version of the constraint in (3): (D++ + iV ++)v+ = 0 ⇒ v+ = φαu+ α + θ+ω1 + θ̄+ω̄2 − 2iθ+θ̄+(φ̇α + iBφα)u−α . This multiplet (“spin multiplet”) is an important ingredient of SQM models with non-Abelian background gauge fields (Sections 4 and 5). 5. Some other multiplets. Of use in the N = 4 SQM model-building are also a fermionic counterpart (0,4,4) of the multiplet (4,4,0), as well as the multiplet (1,4,3) and the chiral multiplet (2,4,2). The first multiplet is described by a fermionic analog Ψ+A(ζ, u) of the superfield q+a, with the harmonic constraint D++Ψ+A = 0 ⇒ Ψ+A = ψαAu+ α + θ+dA + θ̄+d̄A − 2iθ+θ̄+ψ̇αAu−α , (6) and the free action Sfree(Ψ) ∼ ∫ dtAdudθ +dθ̄+ Ψ+AΨ+ A ∼ ∫ dt ( iψαAψ̇αA − dAd̄A ) . The multiplets (1,4,3) and (2,4,2) can be also described within the harmonic superspace setting, though in a rather indirect manner [13, 14]. Most of the analytic N = 4 multiplets listed here have their nonlinear counterparts, with the nonlinearly modified harmonic constraints. Their implications in the N = 4 SQM models have not yet been fully explored so far. Also, in accordance with the fact that the full automorphism group of N = 4, d = 1 superalgebra is SO(4) ∼ SU(2)×SU(2), each N = 4 supermultiplet from the above list has its “mirror” (or “twisted”) counterpart, with the two SU(2) automorphism groups switching their roles. The free actions of all these N = 4 multiplets can be generalized to involve a non-trivial self- interaction. The corresponding bosonic manifolds exhibit interesting target space geometries. Harmonic Superfields in N = 4 Supersymmetric Quantum Mechanics 5 2.3 Bi-harmonic superfields A unified description of N = 4 supermultiplets and their mirror cousins is achieved in the frame- work of bi-harmonic N = 4, d = 1 HSS [25], with the two independent sets of harmonic variables u±1α, v±1i, u1αu−1 α = 1, v1iv−1 i = 1, for either two mutually commuting SU(2) automorphism groups of N = 4, d = 1 supersymmetry. In this approach, the N = 4, d = 1 spinor derivatives are combined into the SU(2)× SU(2) quartet, Dαi = ( Dα, D̄α ) , and then are split into a set of bi-harmonic projections Dαi ⇒ ( D1,1, D1,−1, D−1,1, D−1,−1 ) , where D±1±1 = Dαiu±1 α v±1 i , etc. In this language, the standard harmonic analytic superfields discussed in the previous subsec- tions are defined by the constraints (a) D1,1Φ(I,0) = D1,−1Φ(I,0) = 0, (b) D0,2Φ(I,0) = 0, (7) where I is the harmonic charge with respect to the u-harmonics and D0,2 is the analyticity- preserving covariant derivative with respect to the v-harmonics (the harmonic constraint (7b) just eliminates the v-dependence in the central basis). The mirror multiplets are represented by the alternative analytic superfields Φ(0,J): (a) D1,1Φ(0,J) = D−1,1Φ(0,J) = 0, (b) D2,0Φ(0,J) = 0, with D2,0 being the same as D++ defined above and J the harmonic charge associated with the v-harmonics. These two types of N = 4, d = 1 harmonic analyticity conditions cannot be imposed on the bi-harmonic superfields simultaneously, since {D1,−1, D−1,1} ∼ ∂t . One of the advantages of the bi-harmonic approach is that it makes manifest both SU(2) automorphism groups in their realization on the component fields. For instance, two mutually mirror (4,4,0) multiplets amount to the following sets of d = 1 fields: (xαa, χia) and (xia ′ , χαa ′ ), the first set being another form of the multiplet (3). Analogously, there are two versions of the fermionic off-shell multiplet (0,4,4), (ψαA, diA) and (ψiA ′ , dαA ′ ). The bi-harmonic formalism also helps to establish direct relations between various N = 4, d = 1 multiplets and their twisted cousins. Consider, e.g., a twisted version Ψ(0,1)A′ of the fermionic multiplet (6): D1,1Ψ(0,1)A′ = D−1,1Ψ(0,1)A′ = 0, D2,0Ψ(0,1)A′ = D0,2Ψ(0,1)A′ = 0 ⇒ Ψ(0,1)A′ ∝ ( ψiA ′ , dαA ′) . Then the bosonic superfield Q(1,0)A′ ≡ D1,−1Ψ(0,1)A′ satisfies the standard u-type harmonic analyticity constraints (2), (3) D1,1Q(1,0)A′ = D1,−1Q(1,0)A′ = 0, D2,0Q(1,0)A′ = D0,2Q(1,0)A′ = 0, and so it is a “composite” version of the (4,4,0) multiplet, Q(1,0)A′ ∝ ( dαA ′ , χiA ′ ∼ ψ̇iA′ ) . The invariants (4) and (5), upon substitution q+a ≡ q(1,0)a ⇒ Q(1,0)A′ , would produce some non- minimal actions for the multiplet Ψ(0,1)A′ with non-canonical numbers of time derivatives on the fermionic field ψiA ′ ; in particular, the WZ-type invariant (5) would contain the term with two derivatives ∼ ∫ dtψ̇iA ′ ψ̇B ′ i C(A′B′). For simplicity, in the subsequent consideration we shall stick to the standard N = 4, d = 1 HSS with one set of harmonic variables u±α. 6 E.A. Ivanov 3 Gauging in N = 4, d = 1 HSS The N = 4, d = 1 superfield gauging procedure has been worked out in [12, 13, 14]. It allows one to relate various N = 4 multiplets and their invariant superfield actions with preserving, at each step, the manifest N = 4, d = 1 supersymmetry. 3.1 A simple example of d = 1 gauging in bosonic system Let us start with a simple clarifying bosonic example. Consider a complex d = 1 field z(t), z̄(t) with the following Lagrangian: L0 = ż ˙̄z + iκ (żz̄ − z ˙̄z) . (8) The first term is the kinetic energy, the second one is the simplest d = 1 WZ term. One of the symmetries of this system is the invariance under U(1) transformations: z′ = e−iλz, z̄′ = eiλz̄. Now we gauge this symmetry by promoting λ → λ(t). The gauge invariant action involves the d = 1 gauge field A(t) Lgauge = (ż + iAz)( ˙̄z − iAz̄) + iκ (żz̄ − z ˙̄z + 2iAzz̄) + 2cA, A′ = A+ λ̇, where a “Fayet–Iliopoulos term” ∼ c has been also added. This term is gauge invariant (up to a total derivative) by itself. The next step is to choose the appropriate gauge in Lgauge: z = z̄ ≡ ρ(t). We substitute it into Lgauge and obtain: Lgauge = (ρ̇+ iAρ) (ρ̇− iAρ) + 2iκAρ2 + 2cA = (ρ̇)2 +A2ρ2 − 2κAρ2 + 2cA. The field A(t) is the typical example of auxiliary field: it can be eliminated by its algebraic equation of motion: δA : A = κ− c ρ2 . The final form of the gauge-fixed action is as follows Lgauge ⇒ (ρ̇)2 − ( κρ− c ρ )2 . (9) This is a one-particle prototype of the renowned Calogero–Moser system. At κ = 0, one recovers the standard conformal mechanics: L(κ=0) gauge = (ρ̇)2 − c2 ρ2 . This gauging procedure can be interpreted as an off-shell Lagrangian analog of the well known Hamiltonian reduction. In the present case, in the parametrization z = ρeiϕ, the Hamiltonian reduction consists in imposing the constraints pϕ − 2c ≈ 0, ϕ ≈ 0, upon which the Hamiltonian of the system (8) is reduced to the Hamiltonian of (9). Harmonic Superfields in N = 4 Supersymmetric Quantum Mechanics 7 3.2 An example of supersymmetric gauging in N = 4, d = 1 HSS Now we start from the free action of the multiplet (4,4,0), S = ∫ dtd4θdu q+aD−−q+ a . (10) It is invariant under the shifts q+a → q+a + λu+a, a = 1, 2. The gauging of this Abelian symmetry is accomplished by replacing λ→ Λ(ζ, u) and properly covariantizing (3) D++q+a = 0 ⇒ ∇++q+a = D++q+a − V ++u+a = 0, S ⇒ Sg = ∫ dtd4θdu q+a∇−−q+ a , ∇−−q+ a = D−−q+a − V −−u+a, [∇++,∇−−] = D0 ⇒ D++V −− −D−−V ++ = 0, V −− = V −−(V ++, u). As the next step, we choose the gauge u−aq+ a = 0⇒ q+a = u−aL++. Then D++q+a − V ++u+a = 0 ⇒ V ++ = L++, D++L++ = 0, D++V −− −D−−L++ = 0 ⇒ V −− = 1 2 (D−−)2L++, Sg = ∫ dtd4θduV −−L++ = 1 2 ∫ dtd4θduL++(D−−)2L++. Thus, starting from the free action of the (4,4,0) multiplet and gauging a symmetry of this action, we have eventually arrived at the free action of the multiplet (3,4,1)! As distinct from the previous example, in the present case the gauging procedure does not produce any interaction of the multiplet (3,4,1). Such an interaction could be induced [12] if we would gauge another Abelian symmetry of the action (10), that with respect to the U(1) transformations δq+a = λCab q +b, where Cab is a constant traceless matrix, Caa = 0. 3.3 Further gaugings The superfield gauging procedure just described can be equally applied to other (4,4,0) La- grangians exhibiting some isometries and involving an interaction, equally as to other N = 4, d = 1 multiplets. These multiplets and their superfield actions can be reproduced as the ap- propriate gaugings of the multiplet (4,4,0) or of some nonlinear generalizations of the latter. Below we give a list of such gaugings: • (4,4,0) ⇒ the linear (3,4,1) multiplet – via gauging of shifting or rotational U(1) sym- metry of q+a; • (4,4,0) ⇒ the non-linear (3,4,1) multiplet – via gauging target space scaling symmetry, q+a′ = λq+a; • (4,4,0) ⇒ the chiral (2,4,2) multiplets – via gauging some two-generator solvable sym- metries realized on q+a; • (4,4,0) ⇒ the (1,4,3) multiplet – via gauging SU(2)PG symmetry, q+a′ = λabq +b; • (4,4,0) ⇒ the fermionic (0,4,4) multiplet – via gauging the semi-direct product of SU(2)PG and the shift symmetry δq+a = λu+a. It is worth noting that the d = 1 gauging procedure outlined here resembles the gauging of isometries by non-propagating gauge fields in d = 2 sigma models, which provides a field- theoretical realization of T -duality [26]. There is an essential difference between the d = 1 and d = 2 cases, however. An important part of the d = 2 procedure is the insertion into the 8 E.A. Ivanov action, with a Lagrange multiplier, the condition that the corresponding gauge field strength is vanishing. No gauge field strength can be defined in d = 1, so no analogous constraint is possible. The gauge field finally becomes just the auxiliary field of another N = 4, d = 1 multiplet, and its actual role is to produce some new potential terms in the on-shell action of the latter. 4 N = 4, 4D SQM models in the gauge field backgrounds 4.1 N = 4, 4D SQM with Abelian external gauge field N = 4 SQM model with coupling of (4,4,0) multiplet to the background Abelian gauge field is described by the action [8]: S = ∫ dtd4θduRkin(q+a, D−−q+b, u) + ∫ dtAdudθ +dθ̄+ L+2(q+a, u) ≡ S1 + S2. (11) The second term in (11) involves only one time derivative on the bosonic field xαa, and so it is an example of d = 1 WZ term S2 ∼ ∫ dt ( Aαb(x)ẋαb + fermions ) , Aαb(x) = ∫ duu−α ∂L+2 ∂q+b ∣∣∣ θ=0 , Fαb βd = ∂αbAβd − ∂βdAαb = εαβF(bd), F(αβ) = 0 (self-duality condition). Thus N = 4 supersymmetry requires the external gauge field to be self-dual5. No such a re- quirement is implied, e.g., by N = 2, d = 1 supersymmetry. How to extend this to the most interesting non-Abelian case? 4.2 Non-Abelian self-dual background The coupling to non-Abelian backgrounds can be accomplished by adding the “spin” multiplet (4,4,0) [20]. The relevant superfield action consists of the three pieces: S = ∫ dtd4θduRkin(q+a, D−−q+b, u)− ik 2 ∫ dtAdudθ +dθ̄+ V ++ − 1 2 ∫ dtAdudθ +dθ̄+K(q+a, u)v+v̄+ ≡ S1 + S2 + S3, where the new spin multiplet superfields obey the constraints: (D++ + iV ++)v+ = (D++ − iV ++)v̄+ = 0. In the total action, the piece S1 describes a sigma-model type interaction of xαa: S1 ∼ ∫ dt ( f−2(x)ẋαaẋαa + fermions ) , f−2(x) ∼ ∫ du2Rkin\|θ=0 − conformal factor. The term S2 is the one-dimensional “Fayet–Iliopoulos” term: S2 = k ∫ dtB. 5The analytic function L+2(xαau+ α , u ±) (prepotential) yields the most general solution of the R4 self-duality constraint in the Abelian case [27, 10]. Harmonic Superfields in N = 4 Supersymmetric Quantum Mechanics 9 The term S3 is most important for our purpose. It is a generalized WZ term: S3 ∼ ∫ dt [ iϕ̄α(ϕ̇α + iBϕα)− 1 2 ϕ̄βϕγ (Aαa) γβ ẋαa + fermions without ∂t ] , where (Aαb) γβ = i h ( εαβ ∂ γ bh− 1 2 δγβ ∂αbh ) , h(x) = ∫ duK(x+a, u±β ), 2h(x) = 0. The background gauge field Aαb is self-dual, Fαβ = 0. It precisely matches with the general ’t Hooft ansatz for 4D self-dual SU(2) gauge fields: (Aαb)βγ ⇒ (Aµ)βγ = 1 2 Aiµ(σi) β γ , Aiµ = −η̄iµν∂ν lnh(x), η̄kij = εkij , η̄k0i = −η̄ki0 = δki, i, j, k = 1, 2, 3. An instructive example is the one-instanton configuration on S4: ds2 = 4R4dx2 µ (x2 +R2)2 , Aiµ = 2R2η̄iµνxν x2(x2 +R2) . It corresponds to the following choice of the functions K(x+a, u) and h(x): K(x+a, u±β ) = 1 + 1( c−a x+a )2 , h(x) = 1 + R2 x2 µ , c−a = cαau−α , R2 = \|c\|−2, and can be brought in the BPST form, Âiµ = 2ηiµνxν x2+R2 , F̂ iµν = − 4R2ηiµν (x2+R2)2 , by the singular gauge transformation Aµ → Âµ = U †AµU + iU †∂µU, U(x) = −iσµxµ/ √ x2. (12) 4.3 N = 4 SQM with Yang monopole As a by-product, our non-Abelian SQM construction solves the long-lasting problem of setting up N = 4 SQM model with Yang monopole as a background. Let us consider the following bosonic Lagrangian: LR5 = 1 2 (ẏ5ẏ5 + ẏµẏµ) + B iµ(y) 1 2 (ϕ̄σiϕ) ẏµ, µ = 1, 2, 3, 4, (13) where B iµ = ηiµνyν r(r + y5) , r = √ y2 5 + y2 µ, is the standard form of the Yang monopole potential [28]. Thus (13) describes a coupling of the non-relativistic particle (y5, yµ) in the 5-dimensional Euclidean space R5 to the external Yang monopole field. After the polar decomposition of R5 into the angular S4 ∼ {ỹµ} and the radial r parts as (y5, yµ) ⇒ ( r, √ 1− ỹ2 µ, ỹµ ) , and passing to the stereographic-projection coordinates as ỹµ = 2 xµ 1 + x2 , 10 E.A. Ivanov we obtain LR5 = 1 2 { ṙ2 + 4r2 ẋµẋµ (1 + x2)2 } + 2ηiµνxν ẋµ 1 2(ϕ̄σiϕ) 1 + x2 . (14) The external gauge field in this Lagrangian is just BPST instanton on S4. Hence, if we set the radial coordinate r in (14) equal to a constant, this Lagrangian can be extended to a particular form of the Lagrangian of N = 4 SQM with the self-dual SU(2) gauge field. Thus the 5D mechanics with the gauge coupling to Yang monopole and “frozen” radial coordinate r admits an extension to N = 4 SQM model. The radial coordinate can presumably be described by the N = 4 supermultiplet (1,4,3) which also admits a description in the N = 4 HSS [13] and so can be properly coupled to the set of the basic analytic superfields q+a, v+ and v̄+. 4.4 Quantization of spin variables The (iso)spin variables ϕα, ϕ̄α play the pivotal role for attaining the N = 4 coupling to the external non-Abelian gauge fields. Let us dwell in some detail on their role in the quantum theory. The relevant part of the total action reads: S = ∫ dt [ iϕ̄α(ϕ̇α + iBϕα) + kB +AiµT i ẋµ ] , T i = 1 2 ϕ̄α ( σi ) β α ϕβ, (15) where k = integer. (16) The condition (16) can be deduced from the requirement of invariance of the Euclidean path integral under topologically non-trivial gauge transformations [29]: B(t) → B(t) + α̇(t), ϕ(t) → e−iα(t)ϕ(t). By varying with respect to the “gauge field” B(t), one obtains the constraint on ϕ, ϕ̄: ϕ̄αϕα = k. (17) Applying the standard Dirac quantization procedure, one is left with the commutation relations: [ϕα, ϕ̄ β] = δβα, [ϕα, ϕβ] = [ϕ̄α, ϕ̄β] = 0, ϕα → ∂/∂ϕ̄α. After quantization, the constraint (17) becomes the condition on the wave function ϕ̄αϕαΨ = ϕ̄α ∂ ∂ϕ̄α Ψ = kΨ. (18) It restricts the wave functions to be homogeneous polynomials of ϕ̄α of degree k. The bilinear combinations of the spin variables T i appearing in (15), after quantization are identified as SU(2) generators: T i → T i = 1 2 ϕ̄α ( σi ) β α ∂ ∂ϕβ , [T i, T k] = iεiklT l. Taking into account the constraint (18), one derives T iT i = 1 4 [ (ϕ̄αϕα)2 + 2(ϕ̄αϕα) ] = k 2 ( k 2 + 1 ) . Thus T i are generators of SU(2) in the irrep of spin k/2. An interesting feature is that this gauge SU(2) group is at the same time the R-symmetry group of N = 4 supersymmetry6. 6The gauge transformation (12) converts this SU(2) into another SU(2) which acts on the extra indices a of xαa and commutes with N = 4 supersymmetry. Harmonic Superfields in N = 4 Supersymmetric Quantum Mechanics 11 5 N = 4, 3D SQM in a non-Abelian monopole background 5.1 Superfield action in HSS We can choose off-shell (3,4,1) multiplet L++(ζ, u) instead of the (4,4,0) one q+a(ζ, u) as the dynamical (co-ordinate) multiplet and still keep the gauged (4,4,0) multiplet v+(ζ, u), v̄+(ζ, u) to represent semi-dynamical spin degrees of freedom. This gives rise to N = 4, 3D SQM with coupling to non-Abelian 3D gauge background [21]. The corresponding total superfield action is S = ∫ dtd4θduRkin(L++, L+−, L−−, u)− ik 2 ∫ dtAdudθ +dθ̄+ V ++ − 1 2 ∫ dtAdudθ +dθ̄+K(L++, u)v+v̄+ ≡ S1 + S2 + S3. The first two pieces produce the kinetic sigma-model type term of the (3,4,1) multiplet and Fayet–Iliopoulos term of the gauge N = 4 multiplet. The third piece describes the WZ-type superfield coupling of the co-ordinate multiplet to the external gauge background. 5.2 Component action For simplicity, we choose the free action for L++, with the Lagrangian ∼ L++(D−−)2L++. The component form of the bosonic part of the full action S is S → ∫ dt [ 1 2 ˙̀2 m +AimT i ˙̀ m + iϕ̄α (ϕ̇α + iBϕα) + kB + 1 8 F 2 + 1 2 F ( U iT i )] , where i = 1, 2, 3, m = 1, 2, 3 and Aim = −εmni∂n lnh, U i = −∂i lnh, T i = 1 2 ϕ̄α ( σi ) β α ϕβ, h(`) = ∫ duK ( `αβu+ αu + β , u ± γ ) , ∆h = 0. The 3D gauge field Aim and potential U i are particular solutions of the Bogomolny equations F imn = εmns∇sU i, with F imn = ∂mAin − ∂nAim + εiklAkmAln, ∇mU i = ∂mU i + εiklAkmU l. 5.3 Quantization and SO(3) example Quantization follows the same line as in the 4D case: [T i, T k] = iεiklT l, T iT i = k 2 ( k 2 + 1 ) . The Hamiltonian, in the case with the free kinetic term for `m, is H = 1 2 (p̂m −Am)2 + 1 2 U2 + fermionic terms, U ≡ U iT i. (19) A new feature of the 3D case is the appearance of the “induced” potential term ∼ U iUkT iT k which is generated as a result of elimination of the auxiliary field F . The system (19) provides a non-Abelian generalization of the N = 4 SQM model pioneered in [30]. 12 E.A. Ivanov As an example of the gauge-field background, let us quote the SO(3) invariant one: hso(3)(`) = c0 + c1 1√ `2 ⇒ Aim = εmni `n `2 c1 c1 + c0 √ `2 , U i = `i `2 c1 c1 + c0 √ `2 . In the limit c0 = 0 the background gauge field becomes Wu–Yang monopole [31]; the N = 4 SQM for this case was earlier constructed in [22] in a different approach7. 6 Summary and outlook Let us summarize the basic contents of this contribution. One of its incentives was to provide more evidence that the N = 4, d = 1 harmonic super- space [8] is a useful tool of constructing and analyzing SQM models with N = 4 supersymmetry. It allows one to construct off-shell invariant actions, to establish interrelations between different multiplets, to reveal the relevant target geometries, and so on. As one of the recent uses of the d = 1 HSS, off-shell N = 4 supersymmetric couplings of the multiplets (4,4,0) and (3,4,1) to the external non-Abelian gauge backgrounds were presented. They essentially exploit the auxiliary (iso)spin (4,4,0) multiplet. The background should be self-dual and be described by the ’t Hooft ansatz or its static 3D reduction. HSS is indispensable for setting up the relevant off-shell actions. It should be noticed that, for the time being, our off-shell superfield construction is limited to the ’t Hooft ansatz and the gauge group SU(2). It is still an open question how to extend it to the SU(N) gauge group and to the general self-dual backgrounds, e.g., to the renowned ADHM one. Surprisingly, the on-shell actions, with all auxiliary fields eliminated, admit a direct extension to the gauge group SU(N) and general self-dual backgrounds [19, 20, 21]. This is attainable at cost of on-shell realization of N = 4 supersymmetry. It is interesting to inquire if it is possible to derive these models from some off-shell superfield approach. We finish by indicating possible applications and further directions of study. It would be interesting to extend our construction of couplings to the external non-Abelian gauge fields to the case of higher N , d = 1 supersymmetries, e.g. to N = 8. Also, an obvious task is to exploit some other N = 4 multiplets to represent the coordinate and/or spin variable sectors, e.g. nonlinear versions of the multiplets (4,4,0) and (3,4,1), the (2,4,2) multiplets, etc. As for applications, it would be tempting to make use of the techniques based on the semi- dynamical spin supermultiplets for the explicit calculations of the world-line superextensions of non-Abelian Wilson loops, with the evolution parameter along the loop as a “time”. One more possible area of using the models constructed and their generalizations includes superextensions of Landau problem and higher-dimensional quantum Hall effect, as well as supersymmetric black-hole stuff. Acknowledgements I thank the Organizers of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design” (Benasque, July 18–30, 2010) for inviting me to participate and for the warm hospitality 7In [22] and in some other works of these authors, it was suggested to describe the spin variables, originally introduced in [4] as a bosonic sector of the multiplet (4,4,0), by the fermionic multiplet (0,4,4). On the road from the superfield action to the component one, these authors make non-canonical replacements of the time derivatives of the fermionic fields by new auxiliary fermionic fields. This procedure basically amounts to the construction of “composite” (4,4,0) multiplet from the (0,4,4) one, as explained in the end of Section 2.3. In view of existence of the direct N = 4 off-shell superfield formulation of the multiplet (4,4,0), the use of the auxiliary fermionic multiplet as the starting point looks artificial and superfluous. 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Fernbach, Interscience, New York, 1969, 345–349. http://dx.doi.org/10.1016/j.physletb.2010.04.048 http://arxiv.org/abs/0910.5162 http://dx.doi.org/10.1007/JHEP05(2010)033 http://arxiv.org/abs/0912.3289 http://dx.doi.org/10.1103/PhysRevD.82.085014 http://arxiv.org/abs/1004.4597 http://dx.doi.org/10.1103/PhysRevD.81.105026 http://arxiv.org/abs/0911.3257 http://dx.doi.org/10.1103/PhysRevD.81.085021 http://arxiv.org/abs/1001.2659 http://dx.doi.org/10.1016/j.aop.2004.08.006 http://arxiv.org/abs/hep-th/0401134 http://dx.doi.org/10.1103/PhysRevD.80.065027 http://arxiv.org/abs/0905.3770 http://dx.doi.org/10.1016/0550-3213(92)90269-H http://arxiv.org/abs/hep-th/9110053 http://dx.doi.org/10.1016/0003-4916(88)90256-4 http://dx.doi.org/10.1063/1.523506 http://dx.doi.org/10.1016/0370-2693(91)90739-D http://dx.doi.org/10.1016/0003-4916(83)90316-0 1 Introduction 2 Harmonic N=4, d=1 superspace 2.1 From the ordinary N=4, d=1 superspace to the harmonic one 2.2 Basic N=4, d=1 multiplets 2.3 Bi-harmonic superfields 3 Gauging in N=4, d=1 HSS 3.1 A simple example of d=1 gauging in bosonic system 3.2 An example of supersymmetric gauging in N=4, d=1 HSS 3.3 Further gaugings 4 N=4, 4D SQM models in the gauge field backgrounds 4.1 N=4, 4D SQM with Abelian external gauge field 4.2 Non-Abelian self-dual background 4.3 N=4 SQM with Yang monopole 4.4 Quantization of spin variables 5 N=4, 3D SQM in a non-Abelian monopole background 5.1 Superfield action in HSS 5.2 Component action 5.3 Quantization and SO(3) example 6 Summary and outlook References

Harmonic Superfields in N=4 Supersymmetric Quantum Mechanics

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